New rotational integrals in space forms, with an application to surface area estimation

A surface area estimator for three-dimensional convex sets, based on the invariator principle of local stereology, has recently motivated its generalization by means of new rotational Crofton-type formulae using Morse theory. We follow a different route to obtain related formulae which are more manageable and valid for submanifolds in constant curvature spaces. As an application, we obtain a simplified version of the mentioned surface area estimator for non-convex sets of smooth boundary.


Introduction
In recent years, new rotational integral formulae have been developed ( [1], [7], [8]). This was motivated by the invariator principle of stereology ( [5], see also [14], p. 285). In these formulae, integration over all r-planes is replaced with invariant integration of a measurement function over all r-planes through a fixed point. In particular, in [5], a convenient method was proposed to estimate the surface area of a convex set from an isotropic plane section (called a pivotal section) through a fixed point in terms of the support function of the pivotal section. Recently, in [15], the pertinent formula ( [5], equation (3.2)) has been generalized to non convex Work was supported by the UJI project P11B2012-24 and the PROMETEOII/2014/062 project. sets, and new rotational Crofton formulae have thereby been developed using Morse theory. Further, in [16], the mentioned new generalization has been adapted to estimate the surface area in three-dimensional microscopy. In the present paper, we follow a different route to obtain related generalizations which are more manageable and valid for submanifolds in space forms of constant curvature. In particular, we arrive at a surface area representation which constitutes a simplification of the analogous formula given in [15], [16]. The subsequent estimation procedure is thereby simplified-the basic task consists in identifying the local maxima and minima of the height function of the pivotal section.

A relation between densities of totally geodesic submanifolds in space forms
Let M n λ denote a simply connected Riemannian manifold of sectional curvature λ (namely, the sphere for λ > 0, the hyperbolic space for λ < 0 and the Euclidean space R n for λ = 0). Further, let L n r denote an r-plane (r n), namely a totally geodesic submanifold of dimension r in M n λ , and let dL n r denote the corresponding density, invariant under the group of Euclidean and non-Euclidean motions.
Let M q be a compact differentiable manifold of dimension q embedded in M n λ . Santaló's formula shows that the volume of M q can be expressed in terms of the integral of the volume of the intersection of M q with all the planes L n r , assuming that r + q n ( [13], equation (14.69)). While the latter formula was obtained for the Euclidean case, in [13], p. 309 (Section 4 of Chapter 17), Santaló shows that the same formula extends to M n λ without any change. An r-plane through a fixed point O in M n λ , and its invariant density, are denoted by L n r[0] and dL n r[0] , respectively ( [13], [12]). One of the first problems considered in integral geometry was to determine the densities dL n r and dL n r [0] . For lines in R 2 the invariant density dL 2 1 was proposed by Crofton (see [4]), whereas the invariant densities for lines and planes in M 3 λ were obtained by Cartan [3] in 1896. In 1935, Blaschke [2] introduced the density dL n r for R n . Following Blaschke's methodology, Petkantschin [11] and Santaló [13] obtained the corresponding densities for λ > 0 and λ < 0, respectively.
In this section, we generalize the expressions obtained in [7] for the density of r-planes in M n λ in terms of the density dL n p[0] of p-planes through a fixed point O, of the density dL p r of r-planes in L n p[0] , and of the distance ̺ from O to L p r . For the Euclidean case, the result can be deduced from equation (49) in [11]. From the new expression, new rotational versions of Santaló's formula, and equivalences among them, are obtained in Section 3. Henceforth, the following notation will be used: The density dL n r of r-planes in M n λ can be expressed in terms of the distance ̺ from a fixed point O to L n r , of the density dL n r+1[0] of an (r + 1)-plane through O containing L n r , and of the density dL r+1 r for r-planes contained in L n r+1[0] , as follows ( [7], Corollary 3.1): Theorem 2.1. For r ∈ {0, 1, . . . , n − 2} and p ∈ {r + 1, r + 2, . . . , n}, the following relation between densities is satisfied: where dL n p[r+1] denotes the density for p-planes in M n λ that contain the fixed (r + 1)plane L n r+1[0] . P r o o f. As justified in [13], p. 309, before equation (17.55), from the expressions of the densities of planes in M n λ it follows that some density decompositions (such as [13], equation (12.53)) have the same form regardless of the sign of λ. Therefore, multiplying both sides of equation (2.2) by dL n p[r+1] , and bearing equation (12.53) of [13] in mind, we obtain Applying again (2.2), we get the result.
Note that (2.2) is a special case of (2.3) for p = r + 1.
Corollary 2.2. The density dL n r satisfies the following identity: where dL n p[p−r−1] denotes the density for p-planes in M n λ that contain a fixed plane of dimension p − r − 1 through O.
P r o o f. As justified in [13], p. 309, the densities given in (12.26) and (12.27) of [13] also hold in M n λ . Therefore,

Rotational versions of Santaló's formula
Let M q be a compact differentiable manifold of dimension q embedded in M n λ . Assume that r + q n and consider the set of r-planes in M n λ . Santaló's formula (equation (14.69) of [13], which is valid in M n λ ), states that where σ k denotes k-dimensional volume, and O k = 2π (k+1)/2 /Γ((k + 1)/2) is the surface area of the k-dimensional unit sphere.
In this paper, we focus on the special case r + q = n. Then equation (14.70) of [13], which is valid in M n λ , gives where N denotes number, so that N (M n−r ∩ L n r ) denotes the number of points of the intersection M n−r ∩ L n r . Next we apply Theorem 2.1 to obtain new rotational formulae from (3.2).
Thus, first we fix L n p[0] and integrate with respect to dL p r over all planes in L p r which intersect M n−r p , and then we integrate with respect to dL n p[0] over all planes is a generic p-plane, is also of class C k (and dimension p − r). The need to specify the order k of smoothness is explained in [10]. In Theorem 4.1, one has p = r + 1, whereby M n−r is assumed to be of class C 2 .
P r o o f. As explained above, the submanifold M n−r p is of class C p−r+1 and dimension p − r, whereby M n−r p ∩ L p r is the set of intersection points. Now, bearing in mind equation (12.36) of [13], namely, and making use of equation (2.3) in equation (3.2), the result follows.
The next corollary shows that for the special case p = r + 1, equation (3.3) yields equation (15) from [7] with q = n − r.
. Then σ n−r (M n−r ) admits a rotational expression which may be obtained directly by substituting p = r + 1 on the right-hand side of equation (3.3), namely, Therefore, equation (3.3) can be written as where, by virtue of Theorem A.1 from [8], . Now, making use in equation (3.10) of the well-known identity [13] (3.12) and bearing equation (3.5) in mind, the rotational formula (3.6) is obtained.

Morse type representation and geometrical interpretations
In this section, a geometric interpretation of equation (3.7) is given in terms of the critical points of height functions. Of particular interest will be the special case r = 1, whereby M n−1 is a hypersurface, and the rotational formula is obtained by intersecting M n−1 with a 2-plane L n 2[0] . The density dL r+1 r may be decomposed as (see [13]) where du r denotes the surface area element of the r-dimensional unit sphere and c r λ (̺) = (c λ (̺)) r , where c λ (̺) is defined in equation (2.1). Then, for the cases λ = 0 (Euclidean) and λ < 0 (hyperbolic), we may write with γ ur (0) = O and γ ′ (0) = u r is given by γ ur (t) = c λ (t)O + s λ (t)u r . Given u r , let h ur : L n r+1[0] → R be the height function whose level hypersurfaces are just the r-planes L r+1 r perpendicular to the geodesic γ ur (t). Note that in the Euclidean case (λ = 0), this height function coincides with the standard height function considered in [15]. We suppose that the level hypersurface L r+1 r is oriented in such a way that the unit vector ν(p), perpendicular to the level set L r+1 r ⊂ L n r+1[0] at p, is given by ν(p) = grad(h ur )(p)/ grad(h ur )(p) , where grad(h ur ) denotes the gradient of h ur .
We consider the height function h ur | C : C → R, which is generally a Morse function, and apply the Morse theory to h ur | C (cf. [10]).
Let p ∈ C be a critical point of h ur | C along C. Then there is a level hypersurface L r+1 r of h ur which satisfies T p C ⊂ T p L r+1 r . We assume that h ur | C is an excellent Morse function for almost all u r ∈ S r , which means that all of the critical points in the direction u r from O are non-degenerate, and no two of them lie on the same level hypersurface (i.e. they have different critical values).
Note that the critical points of h ur | C and h (−ur) | C coincide, whereas the corresponding critical values are the same but with opposite signs. Because the integrals in Then for λ = 0, (4.5) I n−r−1,r (̺) = ̺ n−r n − r .
For λ = 0 and for any given pair (n, r), the integral I n−r−1,r (̺) may be evaluated explicitly with the aid of a mathematical software package such as Mathematica or Maple TM . In all cases, we set I n−r−1,r (0) = 0.  where m represents the number of critical points of the height function h ur | C corresponding to the direction u r ; and for λ > 0, where L r+1 r denotes the r-plane perpendicular to the geodesic with direction u r from O at a distance ̺ = 1 2 π/ √ λ.
P r o o f. The fact that C is a curve of class C 2 in L n r+1[0] for a generic (r + 1)subspace L n r+1[0] follows from Theorem A.1 of [8]. Consider first the case λ 0. Then (4.2) may be written as with the convention ̺ 0 = 0. Thus, As we have set I n−r−1,r (0) = 0, the second term of the preceding integrand vanishes, and the proposed result is obtained. For the case λ > 0, the proof is similar. Thus, Therefore, and because I n−r−1,r (0) = 0, the result follows.
R e m a r k 4.2. Equations (4.6) and (4.7) are pertinent to the representation of surface area (see the left-hand side of equation (3.6)). Hence, the integrands on the right-hand sides of equations (4.6) and (4.7) depend on the scanning direction u r in general. Note that the Euler characteristic is of no concern in this paper-we use Morse type concepts, but our route is different.
In order to see the relation between a critical point p and the curvature, we consider a normal coordinate system of L n r+1[0] at p. Let {U ; x 1 , . . . , x r+1 } be a local chart described by the normal coordinate system centered at p; that is, there exist , and a map ϕ : U → V which define the local chart. Let α : I ⊂ R → L n r+1[0] be a parameterization of C with α(0) = p, and set α := β • ϕ, with β(0) = q ∈ R r+1 and ϕ(q) = p.
Then p is a critical point of h ur | C if and only if grad(h ur )(p) ⊥ α ′ (0). Finally, since grad(h ur )(p)/ grad(h ur )(p) = ν(p), the first part of the theorem follows.
To prove the second part we need to determine when d 2 dt 2 t=0 (h ur • ϕ • β) vanishes. Using local coordinates yields The first term on the right-hand side of equation (4.13) is (4.14) hess(h ur )(α ′ (0), α ′ (0)) = g(∇ α ′ (0) grad(h ur ), α ′ (0)) = − |ε(p)|II Lr (α ′ (0)) = 0, where hess denotes the hessian and II Lr is the second fundamental form of L r at p, which is zero, because L r is totally geodesic. As we have considered a normal coordinate system of L n r+1[0] at p, the Christoffel symbols vanish; thus, Further, the coefficients of the First Fundamental Form in normal coordinates are E(p) = G(p) = 1, F (p) = 0, whereby and the second part of the theorem follows.
R e m a r k 4.4. If the curve α is parameterized by arc length, then from [9] we have that where κ 1 > 0 denotes the first curvature of α.  . Moreover, the sign of the geodesic curvature κ g of the curve at a point p is adopted to be positive or negative according to whether the geodesic curvature vector lies to the right or the left of the curve within the geodesic surface L n 2[0] . For λ = 0 the geodesic curvature coincides with the signed curvature κ of the plane curve.

Application: Extension of the invariator method of stereology to estimate surface area
From equation (4.18) we obtain a simplified version of the Morse-type surface area estimator presented in equation (7) from [16]. We simply associate signed indexes to the critical points of the height function, without resorting to the concept of Euler characteristic. Let Y be a compact set with smooth boundary ∂Y in R 3 . From equation (4.18) we have that the surface area of ∂Y can be expressed as follows: where, by virtue of equation (4.19), For each axial direction u ∈ [0, π) in the pivotal plane L 3 2[0] , the pivotal section is scanned entirely from top to bottom by a sweeping straight line parallel to the axis Ou, in search of critical points. Above the axis Ou, the value of the index ε k is +1, or −1, according to whether the kth critical point is a local maximum, or a local minimum. Below the axis Ou, it is convenient to imagine the pivotal section rotated by an angle of 180 • , and then use the same criterion. As a consequence, the factor 1/(n − 1) (equal to 1/2 in this case) on the right-hand side of equation (4.19) does not appear in (5.2). Because the integral of dL 3 2[0] over the unit hemisphere S 2 + is equal to 2π, the combination of the preceding two equations suggests the unbiased estimator of S(∂Y ) from a single pivotal section, and from a single sweeping direction which constitutes the aforementioned modification of equation (7) from [16]. The surface area estimator given by equation (3.2) from [5] corresponds to the case in which Y is a convex set, whereby m = 2. The latter estimator incorporated two mutually perpendicular sweeping directions. In this case, the summation on the right-hand side of equation (5.3) would be replaced with the mean of two summations. In [6], it was shown that two mutually perpendicular sweeping directions yield an accurate surface area estimation for ellipsoidal particles. E x a m p l e 5.1. Fig. 1(a) represents a pivotal section through a smooth particle Y , namely a section produced by an isotropically oriented plane L 3 2[0] through a fixed pivotal point O previously identifiable in the particle (e.g. a nucleolus of a neuron). The axis Ou has been conveniently oriented as horizontal, but it is supposed to be isotropically oriented about O. In Fig. 1(b), the section is the same, but for the sake of illustration, the pivotal point has a different location relative to the particle. A sweeping line moving parallelly to Ou from top to bottom determines four critical points in each case. In Fig. 1(a), the second critical point is a local minimum, whereby ε 2 = −1. The remaining three critical points are local maxima, hence ε 1 = ε 3 = ε 4 = +1. Thus, in this case, The distances to each critical point from the axis Ou have been denoted by {h k } instead of the {̺ k } used in the rest of the paper, in order to match the notation with that adopted in [5], which evokes 'height' measure. In Fig. 1 (b), however, the four critical points are all local maxima, whereby, in this case, (5.5) S(∂X) = 2π(h 2 1 + h 2 2 + h 2 3 + h 2 4 ).
As suggested above, to identify the local maxima and minima below the axis Ou, it is convenient to imagine the section as if turned upside down.